All Questions
6
questions
3
votes
1
answer
75
views
How to prove that this binomial sum remains positive for $x>1$?
Let's say you have this function for real numbers $x>1$, for some positive integer $n \geq 1$
$$
\sum_{k=0}^{\left \lfloor n/2 \right \rfloor} {x \choose 2k+\frac{1-(-1)^n}{2}}
$$
How would you ...
- 633
6
votes
1
answer
194
views
Conjecture: Shallow diagonals in Pascal's triangle form polynomials whose roots are all real, distinct, and in $(-2,2)$.
Here are the first few "shallow diagonals" in Pascal's triangle.
We can use these shallow diagonals to make polynomials. Note that the even degree polynomials have an extra $\pm x$ at the ...
- 25.7k
6
votes
4
answers
182
views
True or false: If you square the coefficients in the expansion of $(x+1)^n$, the resulting polynomial has $n$ distinct real roots.
It seems that if you square the coefficients in the expansion of $(x+1)^n$, the resulting polynomial has $n$ distinct real roots. (I experimented using desmos.com.)
In other words, I am looking for a ...
- 25.7k
4
votes
0
answers
145
views
A rational function with hidden symmetry and alternating poles and zeros.
Upon answering a question about an equivalence of two binomial sums I have noted that a naturally appearing function has some interesting properties.
Consider the function:
$$
f(m,n_1,n_2;z)=\frac{1}{...
- 26.7k
1
vote
1
answer
51
views
Does the expression $\binom {t}{n+1} + \alpha \binom {t}{n}$ simplify?
I've been working on a problem in economics that involves finding the roots of the polynomial
$$ P(i) = \sum_{n=0}^t \bigg[ i^{n+1} \cdot \bigg(\binom {t}{n+1} + \alpha \binom {t}{n} \bigg ) \bigg ]$$...
- 345
3
votes
2
answers
91
views
Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$
Is there an analytic way to obtain the highest root of the polynomial
$x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known.
The other way to ask ...
- 604